Random digit dialing (RDD) can be a very costly methodology for a rare population, but less costly listed samples are not representative and can induce bias. When the list source has less than 30% coverage of the population, the maximum contribution of the listed sample to precision may be severely limited. In a telephone survey of child care for 3-4 year old children in California, we conducted parallel RDD and listed samples and parallel weighted estimation within each sample. We combined these parallel estimates to produce minimum MSE composite estimates for parameters of interest, with weights inversely proportionate to the estimated MSE of each estimate. In particular, we estimated the bias of the listed estimate using the RDD estimate as a gold standard. This approach allows the listed sample to contribute differently to different parameter estimates as a function of estimated bias. We discuss the extent to which this approach improves the MSE of estimates relative to RDD alone and unbiased stratified estimates in the present case.